Ziegler–Nichols Tuning Calculator

A loop that oscillates at K_u = 4 with a 2 min period gets a Ziegler–Nichols PID gain of 2.4, with τ_I = 1 min and τ_D = 0.25 min.

Method:

Closed loop: ultimate gain and periodOpen loop: process reaction curve (FOPDT)

Ultimate gain K_u:

Ultimate period P_u (min):

PID controller gain K_c =

2.4

Precision

PID integral time τ_I: 1 min
PID derivative time τ_D: 0.25 min

Ziegler–Nichols settings
Controller	K_c	τ_I (min)	τ_D (min)
P	2	—	—
PI	1.8	1.66667	—
PID	2.4	1	0.25

Step-by-step solution

Ziegler–Nichols closed loop: P 0.5K_u; PI 0.45K_u, P_u/1.2; PID 0.6K_u, P_u/2, P_u/8

Ziegler–Nichols Tuning Formula

PID: K_c = 0.6K_u, τ_I = P_u/2, τ_D = P_u/8

reaction curve: K_c = 1.2τ/(Kθ), τ_I = 2θ, τ_D = 0.5θ

Worked Example

A loop with ultimate gain 4 and ultimate period 2 min:

Ziegler–Nichols closed loop: P 0.5K_u; PI 0.45K_u, P_u/1.2; PID 0.6K_u, P_u/2, P_u/8

PID derivative time τ_D: 0.25 min

Ziegler–Nichols settings
Controller	K_c	τ_I (min)	τ_D (min)
P	2	—	—
PI	1.8	1.66667	—
PID	2.4	1	0.25

GATE tip

K_u can come from a Routh array or from the frequency where the phase lag is 180°: K_u = 1/|G(jω)| there.

More Instrumentation and Process Control Calculators

First-Order System Response Calculator

y(t) = K M (1 − e^(−t/τ))

G(s) = K/(τs + 1); 63.2 % at t = τ

Example: A process with K = 2 and τ = 3 after a step of 5, at t = 6: Response y(t) = 8.64665.

Open the first-order system response calculator →

Frequently asked questions

How are K_u and P_u found?

With P-only control, raise the gain until the loop oscillates steadily: that gain is K_u and the oscillation period P_u.

Are Ziegler–Nichols settings conservative?

No: they aim for a quarter-decay ratio, which is aggressive; many plants detune them.

Sources

The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .

Last reviewed: 2026-10-15. Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.

Cite this page

APA

Scientific Converters. (2026, October 15). Ziegler–Nichols Tuning Calculator. Retrieved October 29, 2026, from https://scientificconverters.com/calculators/ziegler-nichols-calculator/

IEEE

Scientific Converters, "Ziegler–Nichols Tuning Calculator," scientificconverters.com. Accessed: Oct. 29, 2026. [Online]. Available: https://scientificconverters.com/calculators/ziegler-nichols-calculator/